PONT Mathieu

PhD graduated
Team : APR
    Sorbonne Université - LIP6
    Boîte courrier 169
    Couloir 25-26, Étage 3, Bureau 303
    4 place Jussieu
    75252 PARIS CEDEX 05

Tel: +33 1 44 27 88 16, Mathieu.Pont (at) nulllip6.fr

Supervision : Julien TIERNY

Analysis of Ensembles of Topological Descriptors

Topological Data Analysis (TDA) forms a collection of tools to generically, robustly and efficiently reveal implicit structural patterns hidden in complex datasets. These tools allow computing a topological representation for each member of the ensemble of datasets by encoding its main features of interest in a concise and informative manner. A major challenge consists then in designing analysis tools for such ensembles of topological descriptors. Several tools have been well studied for persistence diagrams, one of the most used descriptors. However, they suffer from a lack of specificity, which can yield identical data representations for significantly distinct datasets
In this thesis, we aimed at developing more advanced analysis tools for ensembles of topological descriptors, capable of tackling the lack of discriminability of persistence diagrams and going beyond what was already available for these objects.
First, we adapt to merge trees, descriptors having a better specificity, the tools already available for persistence diagrams such as distances, geodesics and barycenters. Then, we want to go beyond this notion of the average being the barycenter in order to study the variability within an ensemble of topological descriptors.
We then adapt the Principal Component Analysis framework to persistence diagrams and merge trees, resulting in a dimensionality reduction method that indicates which structures in the ensemble are most responsible for the variability. However, this framework allows only to detect linear patterns of variability in the ensemble. To tackle this we propose to generalize this framework to Auto-Encoder in order to detect non-linear, i.e., more complex, patterns in an ensemble of merging trees or persistence diagrams.
Specifically, we propose a new neural network layer capable of processing natively these objects. We present applications of all this work in feature tracking in a time-varying ensemble, data reduction to compress an ensemble of topological descriptors, clustering to form homogeneous groups in an ensemble, and dimensionality reduction to create a visual map indicating how the data are organized regarding each other in the ensemble.

Defence : 12/01/2023

Jury members :

David Coeurjolly, CNRS [Rapporteur]
Vijay Natarajan, Indian Institut of Science Bengalore [Rapporteur]
Elsa Cazelles, CNRS
Stanley Durrleman, INRIA
Roland Kwitt, University of Salzburg
Gabriel Peyré, CNRS
Katharine Turner, Australian National University
Julien Tierny, CNRS

2021-2024 Publications